Tutorial on the Quaternions of Hamilton

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Tutorial on the Quaternions of Hamilton

Mathematics
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Thank you Michele, I will give it a read! :)
thanks!! :) @Astrophysics

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is this some sort of new language??? lol
no, the quaternions were introduced by W. R. Hamilton in the far 1833 :) lol
lol looks very complicated
no, they are not complicated, please study my definitions first @oleg3321
Very easy to understand, thanks again!
Interesting article. Well explained. Hamilton was Ireland's most famous mathematician . Unfortunately his life was cut short by alcoholism.
Thanks! @welshfella
I remember my calc 3 teacher mentioning those to look at if you got bored.. thanks @Michele_Laino
thanks! :) @DanJS
Thank you @Michele_Laino . I will definitely give it a read
Thanks! :) @arindameducationusc
quaternions aren't a field, they are a division ring
Quaternions are a field @oldrin.bataku
also you should introduce those other basis matrices \(\mathbf i,\mathbf k\) more naturally -- something like this: consider for \(z=a+bi\) we have: $$\begin{align*}a+bi\mapsto \begin{pmatrix}a+bi&0\\0&a-bi\end{pmatrix}&=\begin{pmatrix}a&0\\0&a\end{pmatrix}+\begin{pmatrix}bi&0\\0&-bi\end{pmatrix}\\&=a\begin{pmatrix}1&0\\0&1\end{pmatrix}+b\begin{pmatrix}i&0\\0&-i\end{pmatrix}\\&=a\cdot\mathbf1+b\cdot\mathbf i\end{align*}$$ and then we have that \(\mathbf i\cdot\mathbf j=\mathbf k\)
https://en.wikipedia.org/wiki/Quaternion#Noncommutativity_of_multiplication >The fact that quaternion multiplication is not commutative makes the quaternions an often-cited example of a strictly skew field.
I wrote that formula: \[{\mathbf{ij}} = - {\mathbf{ji}} = {\mathbf{k}}\]
https://en.wikipedia.org/wiki/Division_ring >In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. >Division rings differ from fields only in that their multiplication is not required to be commutative. ... Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.
yes, I know you did, but you introduced \(\mathbf k\) as a matrix first without any motivation. the reason for its existence is that we have four linearly independent matrices \(\mathbf {1,i,j,ij}\) since the multiplication is bilinear, and then we just call \(\mathbf {ij}=\mathbf k\)
I never worked with non-commutative algebra
I meant I never cited non-commutative algebra
? fields have commutative multiplication by definition, whereas the quaternions \(\mathbb H\) are a non-commutative algebra over \(\mathbb R\) and a division ring, not a field
when I wrote a matrix in M2(C) it is supposed that the multiplication is non-commutative since as you know, the multiplication betwen matrices is non-commutative
lol, yes, clearly, and these matrices do not commute, so they cannot form a field. the quaternions are not a field.
:) right
For real man, tuts that are wrong are even worse. This is all simple definition stuff you are getting wrong...Please remove.
More precisely, I have used the term "field", since it is commonly used in mathematics literature, a more appropriate term can be "sfield" which indicate a non-commutative field. I have supposed that, being the non-commutative characteristic evident, there was no need to use the term "sfield"
field is not commonly used in mathematics literature for a skew field, this isn't the 1800s anymore
it's called a division ring, that is the appropriate, correct, and commonly used term for algebraic structures like this

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